Open projections and Murray-von Neumann equivalence

Abstract

We characterize the $C^{\star }$-algebras for which openness of projections in their second duals is preserved under Murray-von Neumann equivalence. They are precisely the extensions of the annihilator $C^{\star }$-algebras by the commutative $C^{\star }$-algebras. We also show that the annihilator $C^{\star }$-algebras are precisely the $C^{\star }$-algebras for which all projections in their second duals are open.

11 pages